Buckling-induced interaction between circular inclusions in an infinite thin plate

Area of Science:

• Materials Science
• Solid Mechanics
• Biophysics

Background:

• Slender artificial materials and biological tissue morphogenesis involve stimulating isolated regions (inclusions).
• Inclusions induce internal stresses, leading to out-of-plane deformation (buckling) for stress relaxation.
• Understanding inclusion interactions is crucial for designing growing structures and tissues.

Purpose of the Study:

• To analyze the interaction between two circular inclusions in an infinite plate using the Föppl-von Kármán model.
• To investigate buckling behavior in regions with narrow transition layers (ℓD ≪ R).
• To compare the energetic favorability of same-direction (up-up) versus opposite-direction (up-down) buckling scenarios.

Main Methods:

• Utilized the Föppl-von Kármán model for theoretical analysis.
• Employed numerical investigations for analyzing close separation regions.
• Defined two distinct separation regions: close separation (ℓ-R∼ℓD) and far separation (ℓ-R≫ℓD).

Main Results:

• Interaction energy exhibits exponential decay in the far separation region (Eint ∝ e^-(ℓ-R)/ℓD).
• Interaction energy shows nonmonotonic behavior in the close separation region.
• The up-down buckling solution is consistently more energetically favorable than the up-up solution.
• Observed symmetry breaking in the up-down scenario within the close separation region.

Conclusions:

• The study elucidates the distinct interaction behaviors of inclusions based on their separation distance.
• The energetically favorable up-down buckling mode demonstrates symmetry breaking at close separations.
• Findings provide insights into the mechanics of buckling in thin structures with multiple inclusions.